4.1 Sampling and Surveys

Transcription

1 4.1 Sampling and Surveys Suppose we want to find out what percent of young drivers in the United States text while driving. To answer the question, we will interview 16- to 20- year- olds who live in the United States and drive. Ideally, we would ask them all (take a census). But contacting every driver in this age group wouldn t be practical: it would take too much time and cost too much money. Instead, we put the question to a sample chosen to represent the entire population of young drivers. Population - The population in a statistical study is the entire group of individuals about which we want information. Sample - A sample is the part of the population from which we actually collect information. We use information from a sample to draw conclusions about the entire population. The distinction between population and sample is basic to statistics. To make sense of any sample result, you must know what population the sample represents. Here s an example that illustrates this distinction and also introduces some major uses of sampling. Example Samping Hardwood and Humans Populations and Samples PROBLEM: Identify the population and the sample in each of the following settings. (a) A furniture maker buys hardwood in large batches. The supplier is supposed to dry the wood before shipping (wood that isn t dry won t hold its size and shape). The furniture maker chooses five pieces of wood from each batch and tests their moisture content. If any piece exceeds 12% moisture content, the entire batch is sent back. (b) Each week, the Gallup Poll questions a sample of about 1500 adult U.S.residents to determine national opinion on a wide variety of issues.

2 4.1.1 The Idea of a Sample Survey We often draw conclusions about a whole population on the basis of a sample. Have you ever tasted a sample of ice cream and ordered a cone if the sample tastes good? Since ice cream is fairly uniform, the single taste represents the whole. Choosing a representative sample from a large and varied population (like all young U.S. drivers) is not so easy. The first step in planning a sample survey Sample survey is to say exactly what population we want to describe. The second step is to say exactly what we want to measure, that is, to give exact definitions of our variables. We reserve the term sample survey for studies that use an organized plan to choose a sample that represents some specific population, like the pieces of hardwood and the U.S. adults in the previous example. By our definition, the population in a sample survey can consist of people, animals, or things. Some people use the terms survey or sample survey to refer only to studies in which people are asked one or more questions, like the Gallup Poll of the last example. We ll avoid this more restrictive terminology. Example How Does the Current Population Survey Work? A sample Survey One of the most important government sample surveys in the United States is the monthly Current Population Survey (CPS). The CPS contacts about 60,000 households each month. It produces the monthly unemployment rate and much other economic and social information. To measure unemployment, we must first specify the population we want to describe. The CPS defines its population as all U.S. residents (legal or not) 16 years of age and over who are civilians and are not in an institution such as a prison. The unemployment rate announced in the news refers to this specific population. What does it mean to be unemployed? Someone who is not looking for work for example, a full- time student should not be called unemployed just because she is not working for pay. If you are chosen for the CPS sample, the interviewer first asks whether you are available to work and whether you actually looked for work in the past four weeks. If not, you are neither employed nor unemployed you are not in the labor force. If you are in the labor force, the interviewer goes on to ask about employment. If you did any work for pay or in your own business during the week of the survey, you are employed. If you worked at least 15 hours in a family business without pay, you are employed. You are also employed if you have a job but didn t work because of vacation, being on strike, or other good reason. An unemployment rate of 9.7% means that 9.7% of the sample was unemployed, using the exact CPS definitions of both labor force and unemployed. The final step in planning a sample survey is to decide how to choose a sample from the population. Let s take a closer look at some good and not- so- good sampling methods.

3 4.1.2 How to Sample Badly How can we choose a sample that we can trust to represent the population? The easiest but not the best sampling method just chooses individuals who are close by. Suppose we re interested in finding out how long students at a large high school spent on homework last week, for example. We might go to the school library and ask the first 30 students we see about their homework time. A sample selected by taking the members of the population that are easiest to reach is called a convenience sample. Convenience samples often produce unrepresentative data. Convenience Sample - Choosing individuals who are easiest to reach results in a convenience sample. What s wrong with our convenience sample of students in the library? It s unlikely that this sample represents the homework habits of all students at the school well. In fact, we d expect the sample to overestimate the average homework time in the population since students who hang out in the library might tend to be more studious. This is bias: using a method that will consistently overestimate or underestimate the value you want to know. Bias - The design of a statistical study shows bias if it systematically favors certain outcomes. AP EXAM TIP If you re asked to describe how the design of a study leads to bias, you re expected to identify the direction of the bias. Suppose you were asked, Explain how using a convenience sample of students in your statistics class to estimate the proportion of all high school students who own a graphing calculator could result in bias. You might respond, This sample would probably include a much higher proportion of students with a graphing calculator than in the population at large because a graphing calculator is required for the statistics class. That is, this method would probably lead to an overestimate of the actual population proportion. Convenience samples are almost guaranteed to show bias. So are voluntary response samples, in which people decide whether to join the sample in response to an open invitation. Call- in, write- in, and many Internet polls rely on voluntary response samples. Unfortunately, people who choose to participate are usually not representative of any clearly defined population. Voluntary response samples attract people who feel strongly about the issue in question, often in the same direction. That leads to bias. Voluntary Response Sample - A voluntary response sample consists of people who choose themselves by responding to a general appeal. Voluntary response samples show bias because people with strong opinions (often in the same direction) are most likely to respond. Write- in and call- in opinion polls are almost sure to lead to strong bias. In fact, only about 15% of the public has ever responded to a call- in poll, and these tend to be the same people who call radio talk shows. That s not a representative sample of the population as a whole.

4 Example Illegal Immigrants and Driver s Licenses Online polls Former CNN commentator Lou Dobbs doesn t like illegal immigration. One of his shows was largely devoted to attacking a proposal to offer driver s licenses to illegal immigrants. During the show, Mr. Dobbs invited his viewers to go to loudobbs.com to vote on the question Would you be more or less likely to vote for a presidential candidate who supports giving drivers licenses to illegal aliens? The result: 97% of the 7350 people who voted by the end of the show said Less likely. PROBLEM: What type of sample did Mr. Dobbs use in his poll? Explain how this sampling method could lead to bias in the poll results.

5 CHECK YOUR UNDERSTANDING For each of the following situations, identify the sampling method used. Then explain how the sampling method could lead to bias. 1. A farmer brings a juice company several crates of oranges each week. A company inspector looks at 10 oranges from the top of each crate before deciding whether to buy all the oranges. 2. The ABC program Nightline once asked whether the United Nations should continue to have its headquarters in the United States. Viewers were invited to call one telephone number to respond Yes and another for No. There was a charge for calling either number. More than 186,000 callers responded, and 67% said No.

6 4.1.3 How to Sample Well: Random Sampling In a voluntary response sample, people choose whether to respond. In a convenience sample, the interviewer makes the choice. In both cases, personal choice produces bias. The statistician s remedy is to allow impersonal chance to choose the sample. A sample chosen by chance rules out both favoritism by the sampler and self- selection by respondents. Random sampling, the use of chance to select a sample, is the central principle of statistical sampling. In everyday life, some people use the word random to mean haphazard, as in, That s so random. In statistics, random means due to chance. Don t say that a sample was chosen at random if a chance process wasn t used to select the individuals. The simplest way to use chance to select a sample is to place names in a hat (the population) and draw out a handful (the sample). This is the idea of a simple random sample. Simple Random Sample - A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected. An SRS not only gives each individual an equal chance to be chosen but also gives every possible sample an equal chance to be chosen. There are other random sampling methods that give each individual, but not each sample, an equal chance. For now, let s think about how to actually select an SRS. When you think of an SRS, picture drawing names from a hat to remind yourself that an SRS doesn t favor any part of the population. That s why an SRS is a better method of choosing samples than convenience or voluntary response samples. But writing names on slips of paper and drawing them from a hat doesn t work as well if the population is large. Think about the Current Population Survey, for example, which must draw a sample of size 60,000 every month from the population of over 117 million U.S. households. In practice, people use random numbers generated by a computer or calculator to choose samples. If you don t have technology handy, you can use a table of random digits. Table of Random Digits - A table of random digits is a long string of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with these properties: Each entry in the table is equally likely to be any of the 10 digits 0 through 9. The entries are independent of each other. That is, knowledge of one part of the table gives no information about any other part.

7 Always use the shortest labels that will cover your population. For instance, you can label up to 100 individuals with two digits: 01, 02,, 99, 00. As standard practice, we recommend that you begin with label 1 (or 01 or 001, as needed). Reading groups of digits from the table gives all individuals the same chance to be chosen because all labels of the same length have the same chance to be found in the table. For example, any pair of digits in the table is equally likely to be any of the 100 possible labels 01, 02,, 99, 00.Ignore any group of digits that was not used as a label or that duplicates a label already in the sample. You can read digits from a table of random digits in any order across a row, down a column, and so on because the table has no order. We recommend reading across rows from left to right. Example Spring Break Choosing an SRS with Table D (table of random digits) The school newspaper is planning an article on family- friendly places to stay over spring break at a nearby beach town. The editors intend to call 4 randomly chosen hotels to ask about their amenities for families with children. They have an alphabetized list of all 28 hotels in the town. PROBLEM: Use Table D at line 130 to choose an SRS of 4 hotels for the editors to call. SOLUTION: We ll use the two- step process for selecting an SRS using Table D. Step 1: Label. Two digits are needed to label the 28 resorts. We have added labels 01 to 28 to the alphabetized list of resorts below. Step 2: Table. To use Table D, start at the left- hand side of line 130 and read two- digit groups. Skip any groups that aren t between 01 and 28, as well as any repeated groups. Continue until you have chosen four resorts. Here is the beginning of line 130: The first 10 two- digit groups are We skip 5 of these 10 groups because they are too high (over 28) and 2 because they are repeats (both 17s). The hotels labeled 05, 16, and 17 go into the sample. We need one more hotel to complete the sample. The remaining 10 two- digit groups in line 130 are

8 Our SRS of 4 hotels for the editors to contact is: 05 Beach Castle, 16 Radisson, 17 Ramada, and 20 Sea Club. As you saw in the previous example, using Table D to select an SRS can be time- consuming. Your calculator can do even better. (Learn Choosing an SRS on your calculator). There is a difference between sampling without replacement and sampling with replacement. Sampling with replacement will select the same number more than once in a given sample. To deal with this problem, you can generate additional random numbers as needed to replace any repeats. Or you can use a method other than randint (like randsamp on the TI- Nspire) to sample without replacement. Refer to your device s reference manual for more detailed instrutions.

9 4.1.4 Other Sampling Methods The basic idea of sampling is straightforward: take an SRS from the population and use your sample results to gain information about the population. Unfortunately, it s usually very difficult to actually get an SRS from the population of interest. Imagine trying to get a simple random sample of all the batteries produced in one day at a factory. Or an SRS of all U.S. high school students. In either case, it s just not practical to choose an SRS. For starters, it would be difficult to obtain an accurate list of the population from which to draw the sample. It would also be very time- consuming to collect data from each individual that s randomly selected. Sometimes, there are also statistical advantages to using more complex sampling methods. One of the most common alternatives to an SRS involves sampling important groups (called strata) within the population separately. Then these separate sub- samples are combined to form one stratified random sample. This method works best when the individuals within each stratum are similar to one another in a way that affects the variable being measured but there are large differences between strata. Stratified Random Sample - To select a stratified random sample, first classify the population into groups of similar individuals, called strata. Then choose a separate SRS in each stratum and combine these SRSs to form the full sample. Choose the strata based on facts known before the sample is taken. For example, in a study of sleep habits on school nights, the population of students in a large high school might be divided into freshman, sophomore, junior, and senior strata. Unfortunately, stratified random sample has the same initials as simple random sample. Only a simple random sample gets abbreviated SRS, however. Example Who Wrote That Song? Using a Stratified Random Sample A radio station that broadcasts a piece of music owes a royalty to the composer. The organization of composers (called ASCAP) collects these royalties for all its members by charging stations a license fee for the right to play members songs. ASCAP has four million songs in its catalog and collects $435 million in fees each year. How should ASCAP distribute this income among its members? By sampling: ASCAP tapes about 60,000 hours from the 53 million hours of local radio programs across the country each year. Radio stations are stratified by type of community (metropolitan, rural), geographic location (New England, Pacific, etc.), and the size of the license fee paid to ASCAP, which reflects the size of the audience. In all, there are 432 strata. Tapes are made at random hours of randomly selected members of each stratum. The tapes are reviewed by experts who can recognize almost every piece of music ever written, and the composers are then paid according to their popularity.

10 Although a stratified random sample can sometimes give more precise information about a population than an SRS, both sampling methods are hard to use when populations are large and spread out over a wide area. In that situation, we d prefer a method that selects groups of individuals that are near one another. That s the idea of a cluster sample. Cluster Sample - To take a cluster sample, first divide the population into smaller groups. Ideally, these clusters should mirror the characteristics of the population. Then choose an SRS of the clusters. All individuals in the chosen clusters are included in the sample. Imagine a large high school that assigns its students to homerooms alphabetically by last name. The school administration is considering a new schedule and would like student input. Administrators decide to survey 200 randomly selected students. It would be hard to track down an SRS of 200 students, so the administration opts for a cluster sample of homerooms. The principal (who knows some statistics) takes a simple random sample of 8 homerooms and gives the survey to all 25 students in each homeroom. Cluster samples are often used for practical reasons, as in the school survey example.they don t offer the statistical advantage of better information about the population that stratified random samples do. That s because clusters are often chosen for ease or convenience, so they may have as much variability as the population itself. Be sure you understand the difference between strata and clusters. We want each stratum to contain similar individuals, and for there to be large differences between strata. For a cluster sample, we d like each cluster to look just like the population, but on a smaller scale. In a cluster sample, some people take an SRS from each cluster rather than including all members of the cluster. You might say that strata are ideally similar within, but different between, while clusters are ideally different within, but similar between.

11 Example Sampling at a School Assembly Strata or clusters? The student council wants to conduct a survey during the first five minutes of an all- school assembly in the auditorium about use of the school library. They would like to announce the results of the survey at the end of the assembly. The student council president asks your statistics class to help carry out the survey. PROBLEM: There are 800 students present at the assembly. A map of the auditorium is shown below. note that students are seated by grade level and that the seats are numbered from 1 to 800. Describe how you would use each of the following sampling methods to select 80 students to complete the survey. (a) Simple random sample (b) Stratified random sample (c) Cluster sample Solution: (a) To take an SRS, we need to choose 80 of the seat numbers at random. Use randint(1,800) on your calculator until 80 different seats are selected. Then give the survey to the students in those seats. (b) The students in the assembly are seated by grade level. Since students library use might be similar within grade levels but different across grade levels,we ll use the grade level seating areas as our strata. Within each grade s seating area, we ll select 20 seats at random. For the 9th grade, use randint(601,800)to select 20 seats. Use randint(401,600) to pick 20 sophomore seats,randint(201,400) to get 20 junior seats, and randint(1,200) to choose 20 senior seats. Give the survey to the students in the selected seats. (c) With the way students are seated, each column of seats from the stage to the back of the auditorium could be used as a cluster. note that each cluster contains students from all four grade levels, so each should represent the population well. Since there are 20 clusters, each with 40 seats, we need to choose 2 clusters at random to get 80 students for the survey. Use randint(1,20) to select two clusters, and then give the survey to all 40 students in each column of seats.

12 Most large- scale sample surveys use multistage samples that combine two or more sampling methods. For example, the Current Population Survey s monthly survey of 60,000 households uses the following design: Stage 1: Divide the United States into 2007 geographical areas called Primary Sampling Units, or PSUs. Select a sample of 756 PSUs. This sample includes the 428 PSUs with the largest population and a stratified random sample of 328 of the others. Stage 2: Divide each PSU selected into smaller areas called neighborhoods. Stratify the neighborhoods using ethnic and other information, and take a stratified random sample of the neighborhoods in each PSU. Stage 3: Sort the housing units in each neighborhood into clusters of four nearby units.interview the households in a random sample of these clusters. Analyzing data from sampling methods more complex than an SRS takes us beyond basic statistics. But the SRS is the building block of more elaborate methods, and the principles of analysis remain much the same for these other methods.

13 CHECK YOUR UNDERSTANDING The manager of a sports arena wants to learn more about the financial status of the people who are attending an NBA basketball game. He would like to give a survey to a representative sample of the more than 20,000 fans in attendance. Ticket prices for the game vary a great deal: seats near the court cost over $100 each, while seats in the top rows of the arena cost $25 each. The arena is divided into 30 numbered sections, from 101 to 130. Each section has rows of seats labeled with letters from A (nearest the court) to ZZ (top row of the arena). 1. Explain why it might be difficult to give the survey to an SRS of 200 fans. 2. Which would be a better way to take a stratified random sample of fans: using the lettered rows or the numbered sections as strata? Explain. 3. Which would be a better way to take a cluster sample of fans: using the lettered rows or the numbered sections as clusters? Explain.

14 4.1.5 Inference for Sampling The purpose of a sample is to give us information about a larger population. The process of drawing conclusions about a population on the basis of sample data is called inference because we infer information about the population from what we know about the sample. Inference from convenience samples or voluntary response samples would be misleading because these methods of choosing a sample are biased. We are almost certain that the sample does not fairly represent the population. The first reason to rely on random sampling is to eliminate bias in selecting samples from the list of available individuals. Still, it is unlikely that results from a random sample are exactly the same as for the entire population. Sample results, like the unemployment rate obtained from the monthly Current Population Survey, are only estimates of the truth about the population. If we select two samples at random from the same population, we will almost certainly choose different individuals. So the sample results will differ somewhat, just by chance. Properly designed samples avoid systematic bias. But their results are rarely exactly correct, and we expect them to vary from sample to sample. Why can we trust random samples? As the previous Activity illustrates, the results of random sampling don t change haphazardly from sample to sample. Because we deliberately use chance, the results obey the laws of probability that govern chance behavior. These laws allow us to say how likely it is that sample results are close to the truth about the population. The second reason to use random sampling is that the laws of probability allow trustworthy inference about the population. Results from random samples come with a margin of error that sets bounds on the size of the likely error. We will discuss the details of inference for sampling later. Margin of Error - Tells how close the estimate tends to be to the unknown parameter in repeated random sampling. One point is worth making now: larger random samples give better information about the population than smaller samples. For instance, in the Activity, you would get a better estimate of the proportion of red chips in the population using random samples of 40 chips than 20 chips. By taking a very large sample, you can be confident that the sample result is very close to the truth about the population. The Current Population Survey contacts about 60,000 households, so we d expect its estimate of the national unemployment rate to be within about 0.1% of the actual population value. Opinion polls that contact 1000 or 1500 people give less precise results we expect the sample result to be within about 3% of the actual population percent with a given opinion. Of course, only samples chosen by chance carry this guarantee. Lou Dobbs s online sample tells us little about overall American public opinion even though 7350 people clicked a response.

15 4.1.6 Sample Surveys: What Can Go Wrong? Random sampling eliminates bias in choosing a sample. But even a large random sample will give a result that differs from the truth about the population. This sampling variability is described by the margin of error that comes with most poll results. So once we see the magic words randomly selected, do we know we have trustworthy information? It certainly beats voluntary response, but not always by as much as we might hope. Sampling in the real world is more complex and less reliable than choosing an SRS from a list of names in a textbook exercise. Most sample surveys are affected by errors in addition to sampling variability. These errors can introduce bias that makes a survey result meaningless. Good sampling technique includes the art of reducing all sources of error. Let s look at the two main sources of errors in sample surveys: sampling errors and nonsampling errors. Sampling errors The margin of error tells us how much sampling variability to expect,and we can control it by choosing the size of our random sample. It doesn t tell us about sampling errors, mistakes made in the process of taking a sample that could lead to inaccurate information about the population. One source of sampling error is the use ofbad sampling methods, such as voluntary response. We can avoid bad methods. Other sampling errors are not so easy to handle. Sampling often begins with a list of individuals from which we will draw our sample. This list is called the sampling frame. Ideally, the sampling frame should list every individual in the population. Because a list of the entire population is rarely available, most samples suffer from some degree of undercoverage. Undercoverage - Undercoverage occurs when some groups in the population are left out of the process of choosing the sample. A sample survey of households, for example, will miss not only homeless people but prison inmates and students in dormitories. An opinion poll conducted by calling landline telephone numbers will miss households that have only cell phones as well as households without a phone. The results of national sample surveys therefore have some bias due to undercoverage if the people not covered differ systematically from the rest of the population. Sampling errors in careful sample surveys are usually quite small. The real problems start when someone picks up (or doesn t pick up) the phone. Now non- sampling errors take over. Nonsampling errors Nonsampling errors are those that can plague even a census. One of the most serious sources of bias in sample surveys is nonresponse, which occurs when a selected individual cannot be contacted or refuses to cooperate. Nonresponse to sample surveys often exceeds 50%, even with careful planning and several callbacks. Because nonresponse is higher in urban areas, most sample surveys substitute other people in the same area to avoid favoring rural areas in the final sample. If the people contacted differ from those who are rarely at home or who refuse to answer questions, some bias remains. For example, retired people may be more likely to respond, which would give their opinions more weight. In a poll about Social Security reform, this could give a misleading impression of the population s views.

16 Nonresponse - Nonresponse occurs when an individual chosen for the sample can t be contacted or refuses to participate. Some students misuse the term voluntary response to explain why certain individuals don t respond in a sample survey. Their idea is that participation in the survey is optional (voluntary), so anyone can refuse to take part. What the students are actually describing is nonresponse. Think about it this way: nonresponse can occur only after a sample has been selected. In a voluntary response sample, every individual has opted to take part, so there won t be any nonresponse. Example The ACS, GSS, and Opinion Polls How bad is nonresponse? The Census Bureau s American Community Survey (ACS) has the lowest nonresponse rate of any poll we know: only about 1% of the households in the sample refuse to respond. The overall nonresponse rate, including never at home and other causes, is just 2.5%. This monthly survey of about 250,000 households replaces the long form that in the past was sent to some households in the every- ten- years national census. Participation in the ACS is mandatory, and the Census Bureau follows up by telephone and then in person if a household doesn t return the mail questionnaire. The University of Chicago s General Social Survey (GSS) is the nation s most important social science survey. The GSS contacts its sample in person, and it is run by a university. Despite these advantages, its most recent survey had a 30% rate of nonresponse. What about opinion polls by news media and opinion- polling firms? We don t know their rates of nonresponse because they won t say. That s a bad sign. The Pew Research Center for the People and the Press imitated a careful random digit dialing survey and published the results: over 5 days, the survey reached 76% of the households in its chosen sample, but because of busy schedules, skepticism and outright refusals, interviews were completed in just 38% of households that were reached. Combining households that could not be contacted with those who did not complete the interview gave a nonresponse rate of 73%. Another type of nonsampling error occurs when someone gives an incorrect response. A systematic pattern of incorrect responses in a sample survey leads to response bias. People know that they should take the trouble to vote, for example, so many who didn t vote in the last election will tell an interviewer that they did. The race or gender of the interviewer can influence responses to questions about race relations or attitudes toward feminism. Answers to questions that ask people to recall past events are often inaccurate because of faulty memory. Have you visited a dentist in the last 6 months? will often draw a Yes from someone who last visited a dentist 8 months ago. Careful training of interviewers and careful supervision to avoid variation among the interviewers can reduce response bias. Good interviewing technique is another aspect of a well- done sample survey. The wording of questions is the most important influence on the answers given to a sample survey. Confusing or leading questions can introduce strong bias, and changes in wording can greatly change a survey s outcome. Even the order in which questions are asked matters.

17 Example How Do Americans Feel about Illegal Immigrants? Question wording matters Should illegal immigrants be prosecuted and deported for being in the U.S. illegally, or shouldn t they? Asked this question in an opinion poll, 69% favored deportation. But when the very same sample was asked whether illegal immigrants who have worked in the United States for two years should be given a chance to keep their jobs and eventually apply for legal status, 62% said that they should. Different questions give quite different impressions of attitudes toward illegal immigrants. Don t trust the results of a sample survey until you have read the exact questions asked. The amount of nonresponse and the date of the survey are also important. Good statistical design is a part, but only a part, of a trustworthy survey. Does the order matter? Ask a sample of college students these two questions: How happy are you with your life in general? (Answers on a scale of 1 to 5) How many dates did you have last month? There is almost no association between responses to the two questions when asked in this order. It appears that dating has little to do with happiness. Reverse the order of the questions, however, and a much stronger association appears: college students who say they had more dates tend to give higher ratings of happiness about life. Asking a question that brings dating to mind makes dating success a big factor in happiness.

18 CHECK YOUR UNDERSTANDING 1. Each of the following is a source of error in a sample survey. Label each as sampling error or nonsampling error, and explain your answers. (a) The telephone directory is used as a sampling frame. (b) The person cannot be contacted in five calls. (c) Interviewers choose people walking by on the sidewalk to interview. 2. A survey paid for by makers of disposable diapers found that 84% of the sample opposed banning disposable diapers. Here is the actual question: It is estimated that disposable diapers account for less than 2% of the trash in today s landfills. In contrast, beverage containers, third-class mail and yard wastes are estimated to account for about 21% of the trash in landfills. Given this, in your opinion, would it be fair to ban disposable diapers? 11 Explain how the wording of the question could result in bias. Be sure to specify the direction of the bias.